Date & Time: Wednesday, 13th March, 2013 : 4.00 -5.00 p.m.

Venue: Ramanujan Hall

Title: From linear to piece wise linear modelling and back. :

Speaker: Prof. Andreas Griewank, Department of Mathematics, Humboldt University

Abstract: It is shown how functions that are defined by evaluation programs involving : the absolute value function $\rabs()$ (besides smooth elementals), can be : approximated locally by piecewise-linear models in the style of algorithmic, : or automatic, differentiation (AD). The model can be generated by a minor : modification of standard AD tools and it is Lipschitz continuous with respect : to the base point at which it is developed. The discrepancy between the : original function, which is {\em piecewise differentiable} and the piecewise : linear model is of second order in the distance to the base point. : Consequently, successive piecewise linearization yields bundle type methods : for unconstrained minimization and Newton type equation solvers. As a third : fundamental numerical task we consider the integration of ordinary : differential equations, for which we examine generalizations of the midpoint : and the trapezoidal rule for the case of Lipschitz continuous right hand : sides. As a by-product of piecewise linearization, we show how to compute at : any base point some generalized Jacobians of the original function, namely : those that are {\em conically active} as defined by Khan and Barton. :